Price, Income and Cross‑Elasticity of Demand – A‑Level Economics (9708)
Learning objectives
- Write down the midpoint (arc) formulae for price elasticity of demand (PED), income elasticity of demand (YED) and cross‑elasticity of demand (XED).
- Explain the economic meaning of the sign and magnitude of each elasticity.
- Identify the main determinants of each elasticity.
- Calculate any of the three elasticities from given data and interpret the result in the context of the Cambridge AS‑Level syllabus.
- Place elasticities within the wider framework of the AS‑Level syllabus – basic economic ideas, demand & supply, government intervention and macro‑economic context.
1. Foundations – the key concepts behind the syllabus
1.1 Scarcity and choice
- Resources are limited but wants are unlimited – societies must decide how to allocate resources.
- Every choice involves an opportunity cost – the value of the next best alternative foregone.
1.2 Factors of production & the production possibility curve (PPC)
- Factors: land, labour, capital, enterprise.
- PPC shows the maximum combinations of two goods that can be produced with full and efficient use of resources.
- Points inside the curve = under‑utilisation; points on the curve = efficient production; points outside = unattainable.
1.3 Classification of goods
| Type of good | Key characteristic (relevant to elasticities) |
|---|
| Normal (necessity) | YED between 0 and 1 |
| Luxury | YED greater than 1 |
| Inferior | YED negative |
| Substitutes | XED positive |
| Complements | XED negative |
2. Demand, Supply and Market Equilibrium
2.1 Demand
- Law of demand: as price falls, quantity demanded rises (ceteris paribus).
- Determinants of demand (shifts of the demand curve):
- Price of the good (movement along the curve)
- Consumer income (normal vs inferior goods)
- Prices of related goods (substitutes & complements)
- Preferences, expectations, number of buyers.
2.2 Supply
- Law of supply: as price rises, quantity supplied rises (ceteris paribus).
- Determinants of supply (shifts of the supply curve):
- Input prices
- Technology
- Number of sellers
- Expectations of future price
- Taxes, subsidies and regulations.
2.3 Market equilibrium, surplus and shortage
- Equilibrium: where the demand and supply curves intersect – the market‑clearing price (P*) and quantity (Q*).
- Surplus: price above equilibrium → quantity supplied > quantity demanded.
- Shortage: price below equilibrium → quantity demanded > quantity supplied.
2.4 Consumer and producer surplus
- Consumer surplus = area between the demand curve and the price line up to the quantity bought.
- Producer surplus = area between the supply curve and the price line up to the quantity sold.
- Both are measures of welfare; a tax or price control changes these areas.
2.5 Price elasticity of supply (PES)
Definition
PES measures the responsiveness of quantity supplied to a change in the good’s own price.
Mid‑point (arc) formula
$$\text{PES}= \frac{\displaystyle\frac{Q_{2}-Q_{1}}{(Q_{1}+Q_{2})/2}}{\displaystyle\frac{P_{2}-P_{1}}{(P_{1}+P_{2})/2}}$$
Interpretation – sign and magnitude
- Both numerator and denominator are positive, so PES is always positive.
- |PES| > 1 – supply is **elastic** (producers can increase output quickly).
- |PES| = 1 – **unit‑elastic**.
- |PES| < 1 – supply is **inelastic** (output changes little when price changes).
Determinants of PES
| Factor | Effect on PES |
|---|
| Availability of spare capacity | More spare capacity → higher PES |
| Time horizon | Long‑run > short‑run (firms can adjust plant, labour) |
| Ease of storing the good | Storable goods → higher PES |
| Complexity of production | Complex, specialised production → lower PES |
Worked example (PES)
A wheat farmer produces 1 200 t when the market price is £150 t⁻¹. After the price rises to £180 t⁻¹, output increases to 1 440 t.
- Percentage change in quantity: $\displaystyle\frac{1\,440-1\,200}{(1\,200+1\,440)/2}= \frac{240}{1\,320}=0.1818\;(18.18\%)$
- Percentage change in price: $\displaystyle\frac{180-150}{(150+180)/2}= \frac{30}{165}=0.1818\;(18.18\%)$
- PES $=0.1818/0.1818=1.0$ → unit‑elastic supply in the short run.
3. Elasticities of Demand
3.1 Why use percentage‑change (elasticity) formulas?
- Express changes relative to the original level – makes comparisons across goods possible.
- The midpoint (arc) method removes the problem of direction‑dependence.
- Elasticities are dimensionless – they can be used in micro‑ and macro‑contexts.
3.2 Price elasticity of demand (PED)
Definition
PED measures the responsiveness of the quantity demanded of a good to a change in its own price.
Mid‑point (arc) formula
$$\text{PED}= \frac{\displaystyle\frac{Q_{2}-Q_{1}}{(Q_{1}+Q_{2})/2}}{\displaystyle\frac{P_{2}-P_{1}}{(P_{1}+P_{2})/2}}$$
Interpretation – sign and magnitude
- Because the law of demand implies an inverse relationship, the calculated value is negative. Most textbooks report the absolute value.
- |PED| > 1 – **elastic** demand.
- |PED| = 1 – **unit‑elastic** demand.
- |PED| < 1 – **inelastic** demand.
- Very elastic: |PED| > 2; very inelastic: |PED| < 0.5.
Determinants of PED
| Factor | Effect on PED |
|---|
| Availability of close substitutes | More substitutes → higher (more elastic) PED |
| Proportion of income spent on the good | Higher proportion → more elastic |
| Definition of the market (broad vs narrow) | Narrow definition (e.g., “Coca‑Cola”) → more elastic |
| Time horizon | Long‑run > short‑run elasticity |
Worked example (PED)
The price of a cinema ticket falls from £10 to £8 and the quantity demanded rises from 1 200 to 1 560 tickets per week.
- ΔQ% $= \frac{1\,560-1\,200}{(1\,200+1\,560)/2}=0.2609$ (26.09 %).
- ΔP% $= \frac{8-10}{(10+8)/2}= -0.2222$ (‑22.22 %).
- PED $=0.2609/(-0.2222)= -1.174$ → |PED| = 1.17.
- Interpretation: demand is **elastic** over this price range.
3.3 Income elasticity of demand (YED)
Definition
YED measures how the quantity demanded of a good responds to a change in consumer income.
Mid‑point formula
$$\text{YED}= \frac{\displaystyle\frac{Q_{2}-Q_{1}}{(Q_{1}+Q_{2})/2}}{\displaystyle\frac{Y_{2}-Y_{1}}{(Y_{1}+Y_{2})/2}}$$
Interpretation – sign and magnitude
- YED > 0 → **normal good**.
• 0 < YED < 1 – **necessity** (quantity rises less than proportionally).
• YED > 1 – **luxury** (quantity rises more than proportionally).
- YED < 0 → **inferior good** (quantity falls as income rises).
Determinants of YED
- Nature of the good (necessity, luxury, inferior).
- Proportion of income spent on the good.
- Availability of close substitutes or higher‑quality alternatives.
- Time horizon – preferences may evolve in the long run.
Worked example (YED)
Weekly sales of organic apples increase from 500 kg to 650 kg when average disposable income rises from £400 to £440.
- ΔQ% $= \frac{650-500}{(500+650)/2}=0.2609$ (26.09 %).
- ΔY% $= \frac{440-400}{(400+440)/2}=0.0952$ (9.52 %).
- YED $=0.2609/0.0952=2.74$.
- Interpretation: YED > 1 → organic apples are a **luxury** (high‑income) good for this group.
3.4 Cross‑elasticity of demand (XED)
Definition
XED measures the responsiveness of the quantity demanded of one good (B) to a change in the price of another good (A).
Mid‑point formula
$$\text{XED}= \frac{\displaystyle\frac{Q_{B2}-Q_{B1}}{(Q_{B1}+Q_{B2})/2}}{\displaystyle\frac{P_{A2}-P_{A1}}{(P_{A1}+P_{A2})/2}}$$
Interpretation – sign and magnitude
- XED > 0 → goods are **substitutes** (price rise in A raises demand for B).
- XED < 0 → goods are **complements** (price rise in A reduces demand for B).
- |XED| > 1 – strong relationship; |XED| < 1 – weak relationship.
Determinants of XED
- Degree of substitutability or complementarity.
- Availability of other alternatives.
- Market definition (broad vs narrow).
- Time horizon – consumers may discover new substitutes in the long run.
Worked example (XED)
The price of coffee rises from £3.00 to £3.60 per cup. In the same period, the quantity of tea sold increases from 2 000 to 2 300 cups per week.
- ΔQ_Tea% $= \frac{2\,300-2\,000}{(2\,000+2\,300)/2}=0.1395$ (13.95 %).
- ΔP_Coffee% $= \frac{3.60-3.00}{(3.00+3.60)/2}=0.1818$ (18.18 %).
- XED $=0.1395/0.1818=0.77$.
- Interpretation: XED is positive, so coffee and tea are **substitutes**, but |XED| < 1 indicates a relatively weak substitution effect.
4. Government Intervention in Micro‑Markets
4.1 Why intervene?
- Market failure (externalities, public goods, information asymmetry).
- Equity concerns – distribution of income and wealth.
- Macro‑economic objectives (inflation, unemployment) that may require micro‑level action.
4.2 Main policy tools
| Tool | Purpose | Typical effect on market |
|---|
| Tax on a good | Reduce consumption of a negative externality or raise revenue | Supply curve shifts left; price to consumers rises, quantity falls; creates tax incidence (partly borne by consumers, partly by producers). |
| Subsidy | Encourage production/consumption of a positive externality | Supply (or demand) curve shifts right; price to consumers falls, quantity rises. |
| Price ceiling | Protect consumers from excessively high prices (e.g., rent control) | Set a maximum price below equilibrium → shortage, black‑market activity. |
| Price floor | Protect producers (e.g., minimum wage, agricultural price support) | Set a minimum price above equilibrium → surplus, possible government purchase. |
| Buffer‑stock scheme | Stabilise volatile markets (e.g., wheat, oil) | Government buys excess (stores) when price falls and sells when price rises, smoothing price fluctuations. |
4.3 Diagrammatic illustration – tax incidence
Show a standard supply and demand diagram with a per‑unit tax shifting the supply curve upward by the tax amount. Highlight the new equilibrium price paid by consumers (Pc) and the price received by producers (Pp), and shade the tax revenue rectangle.
5. Macro‑economics Snapshot (AS Level)
5.1 National income
- GDP = C + I + G + (X‑M) – market value of final goods and services produced within a country in a given period.
- GNI = GDP + net primary income from abroad.
- Real vs nominal – adjust for inflation using a price index (e.g., CPI).
5.2 Circular flow of income
Closed economy: households provide factors of production to firms and receive factor payments (wages, rent, profit). Households spend income on goods & services, generating revenue for firms. In an open economy, add exports, imports, and the foreign sector.
5.3 Aggregate demand (AD) and aggregate supply (AS)
- AD = C + I + G + (X‑M). Downward‑sloping because of the wealth, interest‑rate and exchange‑rate effects.
- Short‑run AS (SRAS) upward‑sloping – prices are sticky; long‑run AS (LRAS) vertical at full‑employment output.
- Shifts:
- AD right: increase in consumption, investment, government spending or net exports.
- SRAS right: fall in input prices, improvement in technology.
- LRAS right: growth in capital stock, labour force, or total factor productivity.
5.4 Economic growth
- Long‑run increase in LRAS – higher potential output.
- Sources: capital accumulation, labour force growth, improvements in technology, human capital development.
5.5 Unemployment
| Type | Definition | Typical cause (macro‑policy relevance) |
|---|
| Frictional | Short‑term job search | Normal in a dynamic labour market. |
| Structural | Mismatch between skills and job requirements | Technology change, sectoral shifts. |
| Classical (real‑wage) | Wages above market‑clearing level | Minimum wages, strong unions. |
| Demand‑deficient (cyclical) | Insufficient aggregate demand | Recessions, weak investment. |
5.6 Inflation
- Measured by the Consumer Price Index (CPI) or the Retail Price Index (RPI).
- Demand‑pull inflation: AD shifts right faster than SRAS.
- Cost‑push inflation: SRAS shifts left due to higher input costs (e.g., oil price shock).
- Built‑in inflation: wage‑price spiral driven by expectations.
6. Government Macro‑policy
6.1 Fiscal policy
- Government spending (G) and taxation (T) directly affect AD.
- Expansionary fiscal policy: increase G or cut T → AD right → higher output & price level (useful to combat unemployment).
- Contractionary fiscal policy: decrease G or raise T → AD left → lower inflation.
- Multiplier effect: ΔY = k · ΔG (or ΔT) where k = 1/(1‑MPC).
6.2 Monetary policy
- Conducted by the central bank (e.g., Bank of England).
- Tools: open‑market operations, policy interest rate, reserve requirements.
- Expansionary: lower interest rates → investment ↑, consumption ↑, AD right.
- Contractionary: raise rates → investment ↓, AD left.
6.3 Supply‑side (structural) policy
- Goal: shift LRAS right → sustainable growth without inflation.
- Measures: improve education & training, deregulation, tax incentives for R&D, infrastructure investment.
6.4 Policy mix table
| Economic problem | Policy mix (short‑run vs long‑run) |
|---|
| High unemployment + low inflation | Expansionary fiscal + expansionary monetary (short‑run); supply‑side reforms (long‑run) |
| High inflation + low unemployment | Contractionary fiscal + contractionary monetary (short‑run); supply‑side to increase LRAS |
| Stagflation | Supply‑side measures combined with careful monetary tightening |
7. Summary table of demand elasticities
| Elasticity |
Mid‑point formula |
Sign & typical interpretation |
| Price Elasticity of Demand (PED) |
$$\displaystyle\frac{\Delta Q/Q_{avg}}{\Delta P/P_{avg}}$$ |
Negative; |PED| > 1 = elastic, =1 = unit‑elastic, <1 = inelastic. Very elastic >2, very inelastic <0.5. |
| Income Elasticity of Demand (YED) |
$$\displaystyle\frac{\Delta Q/Q_{avg}}{\Delta Y/Y_{avg}}$$ |
Positive = normal (0 < YED < 1 necessity; YED > 1 luxury); Negative = inferior. |
| Cross‑Elasticity of Demand (XED) |
$$\displaystyle\frac{\Delta Q_B/Q_{B,avg}}{\Delta P_A/P_{A,avg}}$$ |
Positive = substitutes, Negative = complements; |XED| > 1 strong, <1 weak. |
| Price Elasticity of Supply (PES) |
$$\displaystyle\frac{\Delta Q/Q_{avg}}{\Delta P/P_{avg}}$$ |
Positive; |PES| > 1 = elastic, =1 = unit‑elastic, <1 = inelastic. |
8. Quick practice questions
- PED: Price falls from £20 to £15, quantity rises from 800 to 1 200 units. Calculate PED and state whether demand is elastic, unit‑elastic or inelastic.
- YED: Income rises from £30 000 to £33 000 and gym‑membership demand rises from 150 to 180 per month. Find YED and classify the good.
- XED: The price of butter increases by 10 % and the quantity demanded of margarine rises by 5 %. Compute XED and identify the relationship between the two products.
- PES: A car manufacturer can increase output from 5 000 to 5 500 units when the market price rises from £20 000 to £22 000 per car. Calculate PES.
- Fiscal policy: The government decides to increase spending by £5 bn. If the multiplier is 1.8, what is the expected change in national income?
9. Key points to remember
- Use the midpoint (arc) formula for all elasticities – it gives a direction‑independent, dimensionless measure.
- Interpretation always requires both the sign (law of demand, normal/inferior, substitutes/complements) and the magnitude (elastic vs inelastic, strong vs weak).
- Determinants of each elasticity are rooted in the basic concepts of scarcity, substitution and time.
- Elasticities are central to analysing the impact of price changes, income changes, related‑good changes, taxes, subsidies and broader macro‑policy.
- Understanding the interaction of demand, supply, equilibrium, surplus, and government intervention equips you to answer both micro‑ and macro‑questions in the Cambridge AS‑Level exam.